A maximum principle for graphs

Abstract Let G be an infinite graph decomposing the plane into polygonal regions. We assume that there are at least 3 edges indident each vertex of G, and at least 6 edges incident with each polygonal region. We define the distance between two vertices to be the least number of edges in any path joining them. Now let C be a simple closed path in G bounding a closed set D, and let 0 be any vertex of G. Then the maximum distance from 0 to a vertex of D is not attained at any vertex interior to D. The same conclusion holds with the pair of numbers (3,6) in the hypothesis replaced by (4,4) or (6,3).